Matlab Codgen eig()函数-奇怪的行为 [英] Matlab Codgen eig() function - strange behaviour
本文介绍了Matlab Codgen eig()函数-奇怪的行为的处理方法，对大家解决问题具有一定的参考价值，需要的朋友们下面随着小编来一起学习吧！
问题描述
首先，不要被冗长的文章所迷惑，没有太多代码只是观察结果而已，因此示例矩阵很少.
First, don't be fooled by the long post, there is not a lot of code just an observation of results so there are few example matrices.
这与这个问题有点相关: Matlab Codegen Eig Function -这是Bug吗?
This is a bit related to this question: Matlab Codegen Eig Function - Is this a Bug?
我知道mex/C/C ++翻译了 eig()在MATLAB中使用相同的函数时，函数可能不会返回相同的特征向量，这很好，但是我对获得的结果感到困惑.
I know that mex/C/C++ translated eig() function may not return the same eigenvectors when using the same function in MATLAB and that's fine, but i am puzzled with results I'm getting.
首先，这个简单的示例:
Output
% c = diagonal matrix of eigenvalues
% b = matrix whose columns are the corresponding right eigenvectors
function [ b, c ] = eig_test(a)
[b, c] = eig(a);
end
按[b,c] = eig_test(magic(5))的原样运行将返回:
b =
-0.4472 0.0976 -0.6330 0.6780 -0.2619
-0.4472 0.3525 0.5895 0.3223 -0.1732
-0.4472 0.5501 -0.3915 -0.5501 0.3915
-0.4472 -0.3223 0.1732 -0.3525 -0.5895
-0.4472 -0.6780 0.2619 -0.0976 0.6330
c =
65.0000 0 0 0 0
0 -21.2768 0 0 0
0 0 -13.1263 0 0
0 0 0 21.2768 0
0 0 0 0 13.1263
将其转换为mex函数并运行eig_test_mex(magic(5))会返回:
Translating that to mex function and running eig_test_mex(magic(5)) returns:
b =
0.4472 + 0.0000i 0.0976 + 0.0000i -0.6330 + 0.0000i 0.6780 + 0.0000i -0.2619 + 0.0000i
0.4472 + 0.0000i 0.3525 + 0.0000i 0.5895 + 0.0000i 0.3223 + 0.0000i -0.1732 + 0.0000i
0.4472 + 0.0000i 0.5501 + 0.0000i -0.3915 + 0.0000i -0.5501 + 0.0000i 0.3915 + 0.0000i
0.4472 + 0.0000i -0.3223 + 0.0000i 0.1732 + 0.0000i -0.3525 + 0.0000i -0.5895 + 0.0000i
0.4472 + 0.0000i -0.6780 + 0.0000i 0.2619 + 0.0000i -0.0976 + 0.0000i 0.6330 + 0.0000i
c =
65.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i -21.2768 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i -13.1263 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 21.2768 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 13.1263 + 0.0000i
现在这里的值实际上是相同的(除了第一个向量的唯一区别是-号不同)，但是这些复杂的虚部是从哪里来的?
Now here the values are actually the same (with exception of first vector that differs only in - sign) but from where are these complex imaginary parts coming from?
第二个示例:
使用与上面相同的功能，输入矩阵现在是下面的这个矩阵，而不是magic(5):
Using the same function as above, the input matrix is now this one below instead of magic(5):
input_matrix =
0.0440 -0.0000 0.0486 0.0752 0.0848 0.0881 0.0883 0.0874 0.0856 0.0832 0.0805 0.0775 0.0742 0.0709 0.0676 0.0644 0.0612 0.0580 0.0548 0.0518 0.0487
-0.0000 0 -0.0000 -0.0000 0 -0.0000 -0.0000 0 0 -0.0000 -0.0000 -0.0000 0 0 -0.0000 -0.0000 0 0 0 -0.0000 0
0.0486 -0.0000 0.1253 0.1231 0.1128 0.1028 0.0940 0.0867 0.0803 0.0747 0.0697 0.0651 0.0609 0.0570 0.0535 0.0503 0.0473 0.0446 0.0421 0.0397 0.0375
0.0752 -0.0000 0.1231 0.3049 0.2850 0.2641 0.2454 0.2297 0.2157 0.2034 0.1924 0.1820 0.1723 0.1636 0.1556 0.1482 0.1414 0.1351 0.1292 0.1237 0.1186
0.0848 0 0.1128 0.2850 0.3941 0.3685 0.3451 0.3253 0.3077 0.2921 0.2780 0.2646 0.2521 0.2407 0.2303 0.2207 0.2119 0.2036 0.1959 0.1887 0.1819
0.0881 -0.0000 0.1028 0.2641 0.3685 0.4420 0.4161 0.3943 0.3746 0.3571 0.3413 0.3262 0.3120 0.2990 0.2871 0.2762 0.2661 0.2566 0.2478 0.2396 0.2318
0.0883 -0.0000 0.0940 0.2454 0.3451 0.4161 0.4705 0.4474 0.4266 0.4079 0.3910 0.3748 0.3595 0.3455 0.3326 0.3208 0.3098 0.2996 0.2900 0.2811 0.2726
0.0874 0 0.0867 0.2297 0.3253 0.3943 0.4474 0.4918 0.4701 0.4507 0.4330 0.4160 0.3999 0.3851 0.3716 0.3591 0.3474 0.3366 0.3265 0.3170 0.3080
0.0856 0 0.0803 0.2157 0.3077 0.3746 0.4266 0.4701 0.5067 0.4868 0.4686 0.4511 0.4343 0.4190 0.4049 0.3919 0.3798 0.3686 0.3580 0.3481 0.3387
0.0832 -0.0000 0.0747 0.2034 0.2921 0.3571 0.4079 0.4507 0.4868 0.5182 0.4997 0.4817 0.4645 0.4487 0.4343 0.4209 0.4084 0.3968 0.3859 0.3757 0.3660
0.0805 -0.0000 0.0697 0.1924 0.2780 0.3413 0.3910 0.4330 0.4686 0.4997 0.5269 0.5087 0.4911 0.4751 0.4603 0.4466 0.4339 0.4220 0.4108 0.4003 0.3905
0.0775 -0.0000 0.0651 0.1820 0.2646 0.3262 0.3748 0.4160 0.4511 0.4817 0.5087 0.5317 0.5140 0.4977 0.4827 0.4688 0.4559 0.4438 0.4325 0.4218 0.4117
0.0742 0 0.0609 0.1723 0.2521 0.3120 0.3595 0.3999 0.4343 0.4645 0.4911 0.5140 0.5337 0.5173 0.5021 0.4881 0.4750 0.4628 0.4514 0.4406 0.4304
0.0709 0 0.0570 0.1636 0.2407 0.2990 0.3455 0.3851 0.4190 0.4487 0.4751 0.4977 0.5173 0.5351 0.5199 0.5057 0.4926 0.4803 0.4687 0.4578 0.4475
0.0676 -0.0000 0.0535 0.1556 0.2303 0.2871 0.3326 0.3716 0.4049 0.4343 0.4603 0.4827 0.5021 0.5199 0.5362 0.5220 0.5087 0.4963 0.4847 0.4737 0.4634
0.0644 -0.0000 0.0503 0.1482 0.2207 0.2762 0.3208 0.3591 0.3919 0.4209 0.4466 0.4688 0.4881 0.5057 0.5220 0.5370 0.5237 0.5112 0.4995 0.4885 0.4781
0.0612 0 0.0473 0.1414 0.2119 0.2661 0.3098 0.3474 0.3798 0.4084 0.4339 0.4559 0.4750 0.4926 0.5087 0.5237 0.5376 0.5251 0.5134 0.5023 0.4918
0.0580 0 0.0446 0.1351 0.2036 0.2566 0.2996 0.3366 0.3686 0.3968 0.4220 0.4438 0.4628 0.4803 0.4963 0.5112 0.5251 0.5381 0.5263 0.5152 0.5047
0.0548 0 0.0421 0.1292 0.1959 0.2478 0.2900 0.3265 0.3580 0.3859 0.4108 0.4325 0.4514 0.4687 0.4847 0.4995 0.5134 0.5263 0.5384 0.5273 0.5167
0.0518 -0.0000 0.0397 0.1237 0.1887 0.2396 0.2811 0.3170 0.3481 0.3757 0.4003 0.4218 0.4406 0.4578 0.4737 0.4885 0.5023 0.5152 0.5273 0.5387 0.5281
0.0487 0 0.0375 0.1186 0.1819 0.2318 0.2726 0.3080 0.3387 0.3660 0.3905 0.4117 0.4304 0.4475 0.4634 0.4781 0.4918 0.5047 0.5167 0.5281 0.5389
运行[b,c] = eig_test(input_matrix)返回:
b =
0.0000 -0.0085 -0.0117 -0.0166 -0.0251 -0.0442 0.1334 0.9260 -0.1493 0.0548 -0.0283 0.0382 0.0170 0.0977 0.1285 -0.1697 0.1100 -0.1149 0.0635 0.0881 0.0424
1.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000
-0.0000 0.0020 0.0029 0.0042 0.0067 0.0125 -0.0404 -0.2919 0.0495 -0.0178 0.0219 0.0179 0.1086 0.2330 0.4679 -0.5139 0.4286 -0.3220 0.2241 0.1454 0.0392
0.0000 0.0001 0.0001 0.0002 0.0003 0.0005 -0.0010 -0.0102 -0.0017 -0.0143 -0.0484 -0.1305 -0.2829 -0.4495 -0.4237 0.0949 0.2592 -0.3986 0.4161 0.3145 0.1072
-0.0000 0.0001 0.0002 0.0003 0.0006 0.0005 -0.0076 -0.0267 0.0241 0.0641 0.1763 0.3391 0.4702 0.3538 -0.0858 0.3847 -0.2103 -0.1113 0.3607 0.3698 0.1531
0.0000 0.0001 0.0001 0.0003 -0.0001 0.0044 0.0173 -0.0280 -0.0740 -0.2070 -0.3832 -0.4693 -0.2586 0.1790 0.3224 0.0909 -0.3636 0.1799 0.2003 0.3654 0.1849
0.0000 0.0001 0.0001 -0.0000 0.0029 -0.0168 -0.0758 0.0155 0.2178 0.4038 0.4549 0.1676 -0.2686 -0.2836 0.1914 -0.2398 -0.2029 0.3295 0.0210 0.3288 0.2079
-0.0000 0.0001 0.0000 0.0015 -0.0114 0.0623 0.1958 -0.0857 -0.3987 -0.4488 -0.1154 0.3136 0.2223 -0.2549 -0.1457 -0.2976 0.0628 0.3254 -0.1323 0.2755 0.2255
-0.0000 0.0001 0.0005 -0.0058 0.0419 -0.1658 -0.3782 0.0966 0.4538 0.1163 -0.3266 -0.1826 0.2893 0.0922 -0.2891 -0.1140 0.2566 0.2114 -0.2394 0.2128 0.2385
0.0000 0.0001 -0.0020 0.0224 -0.1169 0.3338 0.4695 -0.1012 -0.1655 0.3142 0.1810 -0.3027 -0.0583 0.2923 -0.1581 0.1266 0.3013 0.0487 -0.2951 0.1468 0.2482
-0.0000 -0.0003 0.0085 -0.0675 0.2572 -0.4725 -0.2826 -0.0124 -0.2658 -0.2251 0.2927 0.0580 -0.2952 0.1782 0.0838 0.2657 0.2098 -0.1089 -0.3025 0.0811 0.2552
0.0000 0.0016 -0.0282 0.1667 -0.4258 0.4018 -0.1483 0.0743 0.2952 -0.2511 -0.1046 0.3011 -0.1748 -0.0861 0.2409 0.2463 0.0445 -0.2242 -0.2699 0.0183 0.2593
-0.0000 -0.0060 0.0796 -0.3248 0.4873 -0.0371 0.3582 -0.0471 0.1556 0.1904 -0.3026 0.1397 0.1135 -0.2564 0.2258 0.1072 -0.1212 -0.2793 -0.2087 -0.0393 0.2613
-0.0000 0.0202 -0.1839 0.4791 -0.2780 -0.3330 -0.0369 -0.0733 -0.2823 0.2701 -0.0569 -0.1694 0.2707 -0.2224 0.0779 -0.0674 -0.2333 -0.2741 -0.1307 -0.0908 0.2618
0.0000 -0.0569 0.3438 -0.4801 -0.1422 0.2530 -0.3217 0.0344 -0.1598 -0.0783 0.2433 -0.2745 0.1829 -0.0400 -0.1011 -0.1994 -0.2669 -0.2193 -0.0462 -0.1354 0.2611
0.0000 0.1366 -0.4999 0.2085 0.3815 0.1931 0.0665 0.0563 0.2298 -0.2886 0.2299 -0.1025 -0.0438 0.1536 -0.2162 -0.2443 -0.2244 -0.1314 0.0362 -0.1730 0.2594
-0.0000 -0.2750 0.5187 0.2125 -0.1234 -0.2906 0.3002 -0.0330 0.2275 -0.0976 -0.0389 0.1510 -0.2225 0.2429 -0.2221 -0.1981 -0.1272 -0.0288 0.1102 -0.2036 0.2569
0.0000 0.4544 -0.2791 -0.3911 -0.2949 -0.1393 -0.0141 -0.0736 -0.1126 0.2031 -0.2494 0.2586 -0.2356 0.1924 -0.1303 -0.0881 -0.0057 0.0716 0.1710 -0.2273 0.2536
-0.0000 -0.5901 -0.1434 0.0927 0.2187 0.2750 -0.2874 -0.0090 -0.2727 0.2455 -0.2023 0.1516 -0.0947 0.0451 0.0094 0.0433 0.1096 0.1561 0.2158 -0.2445 0.2496
0.0000 0.5431 0.4134 0.3143 0.2317 0.1604 -0.0958 0.0563 -0.0617 0.0119 0.0294 -0.0645 0.0962 -0.1161 0.1372 0.1548 0.1950 0.2157 0.2434 -0.2554 0.2451
-0.0000 -0.2286 -0.2287 -0.2288 -0.2283 -0.2265 0.2263 0.0338 0.2163 -0.2220 0.2231 -0.2228 0.2210 -0.2154 0.2075 0.2171 0.2365 0.2458 0.2538 -0.2607 0.2401
c =
-0.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0.0063 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0.0076 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.0089 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0.0105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0.0123 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.0144 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0.0157 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0.0171 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0.0203 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0.0244 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0.0298 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0.0368 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0.0464 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0581 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0762 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1063 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1746 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3324 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0781 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7.1136
运行[b,c] = eig_test_mex(input_matrix)的mex版本会返回:
And running mex version for [b,c] = eig_test_mex(input_matrix) returns:
b=
0.0424+0.0000i 0.0881+0.0000i 0.0635+0.0000i -0.1149+0.0000i 0.11+0.0000i -0.1697+0.0000i 0.1285+0.0000i 0.0977+0.0000i 0.017+0.0000i 0.0382+0.0000i -0.0283+0.0000i 0.0548+0.0000i 0.1493+0.0000i -0.926+0.0000i -0.1334+0.0000i 0.0085+0.0000i -0.0442+0.0000i -0.0117+0.0000i -0.0251+0.0000i -0.0166+0.0000i 0+0.0000i
+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i 0+0.0000i -1+0.0000i
0.0392+0.0000i 0.1454+0.0000i 0.2241+0.0000i -0.322+0.0000i 0.4286+0.0000i -0.5139+0.0000i 0.4679+0.0000i 0.233+0.0000i 0.1086+0.0000i 0.0179+0.0000i 0.0219+0.0000i -0.0178+0.0000i -0.0495+0.0000i 0.2919+0.0000i 0.0404+0.0000i -0.002+0.0000i 0.0125+0.0000i 0.0029+0.0000i 0.0067+0.0000i 0.0042+0.0000i 0+0.0000i
0.1072+0.0000i 0.3145+0.0000i 0.4161+0.0000i -0.3986+0.0000i 0.2592+0.0000i 0.0949+0.0000i -0.4237+0.0000i -0.4495+0.0000i -0.2829+0.0000i -0.1305+0.0000i -0.0484+0.0000i -0.0143+0.0000i 0.0017+0.0000i 0.0102+0.0000i 0.001+0.0000i -0.0001+0.0000i 0.0005+0.0000i 0.0001+0.0000i 0.0003+0.0000i 0.0002+0.0000i 0+0.0000i
0.1531+0.0000i 0.3698+0.0000i 0.3607+0.0000i -0.1113+0.0000i -0.2103+0.0000i 0.3847+0.0000i -0.0858+0.0000i 0.3538+0.0000i 0.4702+0.0000i 0.3391+0.0000i 0.1763+0.0000i 0.0641+0.0000i -0.0241+0.0000i 0.0267+0.0000i 0.0076+0.0000i -0.0001+0.0000i 0.0005+0.0000i 0.0002+0.0000i 0.0006+0.0000i 0.0003+0.0000i 0+0.0000i
0.1849+0.0000i 0.3654+0.0000i 0.2003+0.0000i 0.1799+0.0000i -0.3636+0.0000i 0.0909+0.0000i 0.3224+0.0000i 0.179+0.0000i -0.2586+0.0000i -0.4693+0.0000i -0.3832+0.0000i -0.207+0.0000i 0.074+0.0000i 0.028+0.0000i -0.0173+0.0000i -0.0001+0.0000i 0.0044+0.0000i 0.0001+0.0000i -0.0001+0.0000i 0.0003+0.0000i 0+0.0000i
0.2079+0.0000i 0.3288+0.0000i 0.021+0.0000i 0.3295+0.0000i -0.2029+0.0000i -0.2398+0.0000i 0.1914+0.0000i -0.2836+0.0000i -0.2686+0.0000i 0.1676+0.0000i 0.4549+0.0000i 0.4038+0.0000i -0.2178+0.0000i -0.0155+0.0000i 0.0758+0.0000i -0.0001+0.0000i -0.0168+0.0000i 0.0001+0.0000i 0.0029+0.0000i 0+0.0000i 0+0.0000i
0.2255+0.0000i 0.2755+0.0000i -0.1323+0.0000i 0.3254+0.0000i 0.0628+0.0000i -0.2976+0.0000i -0.1457+0.0000i -0.2549+0.0000i 0.2223+0.0000i 0.3136+0.0000i -0.1154+0.0000i -0.4488+0.0000i 0.3987+0.0000i 0.0857+0.0000i -0.1958+0.0000i -0.0001+0.0000i 0.0623+0.0000i 0+0.0000i -0.0114+0.0000i 0.0015+0.0000i 0+0.0000i
0.2385+0.0000i 0.2128+0.0000i -0.2394+0.0000i 0.2114+0.0000i 0.2566+0.0000i -0.114+0.0000i -0.2891+0.0000i 0.0922+0.0000i 0.2893+0.0000i -0.1826+0.0000i -0.3266+0.0000i 0.1163+0.0000i -0.4538+0.0000i -0.0966+0.0000i 0.3782+0.0000i -0.0001+0.0000i -0.1658+0.0000i 0.0005+0.0000i 0.0419+0.0000i -0.0058+0.0000i 0+0.0000i
0.2482+0.0000i 0.1468+0.0000i -0.2951+0.0000i 0.0487+0.0000i 0.3013+0.0000i 0.1266+0.0000i -0.1581+0.0000i 0.2923+0.0000i -0.0583+0.0000i -0.3027+0.0000i 0.181+0.0000i 0.3142+0.0000i 0.1655+0.0000i 0.1012+0.0000i -0.4695+0.0000i -0.0001+0.0000i 0.3338+0.0000i -0.002+0.0000i -0.1169+0.0000i 0.0224+0.0000i 0+0.0000i
0.2552+0.0000i 0.0811+0.0000i -0.3025+0.0000i -0.1089+0.0000i 0.2098+0.0000i 0.2657+0.0000i 0.0838+0.0000i 0.1782+0.0000i -0.2952+0.0000i 0.058+0.0000i 0.2927+0.0000i -0.2251+0.0000i 0.2658+0.0000i 0.0124+0.0000i 0.2826+0.0000i 0.0003+0.0000i -0.4725+0.0000i 0.0085+0.0000i 0.2572+0.0000i -0.0675+0.0000i 0+0.0000i
0.2593+0.0000i 0.0183+0.0000i -0.2699+0.0000i -0.2242+0.0000i 0.0445+0.0000i 0.2463+0.0000i 0.2409+0.0000i -0.0861+0.0000i -0.1748+0.0000i 0.3011+0.0000i -0.1046+0.0000i -0.2511+0.0000i -0.2952+0.0000i -0.0743+0.0000i 0.1483+0.0000i -0.0016+0.0000i 0.4018+0.0000i -0.0282+0.0000i -0.4258+0.0000i 0.1667+0.0000i 0+0.0000i
0.2613+0.0000i -0.0393+0.0000i -0.2087+0.0000i -0.2793+0.0000i -0.1212+0.0000i 0.1072+0.0000i 0.2258+0.0000i -0.2564+0.0000i 0.1135+0.0000i 0.1397+0.0000i -0.3026+0.0000i 0.1904+0.0000i -0.1556+0.0000i 0.0471+0.0000i -0.3582+0.0000i 0.006+0.0000i -0.0371+0.0000i 0.0796+0.0000i 0.4873+0.0000i -0.3248+0.0000i 0+0.0000i
0.2618+0.0000i -0.0908+0.0000i -0.1307+0.0000i -0.2741+0.0000i -0.2333+0.0000i -0.0674+0.0000i 0.0779+0.0000i -0.2224+0.0000i 0.2707+0.0000i -0.1694+0.0000i -0.0569+0.0000i 0.2701+0.0000i 0.2823+0.0000i 0.0733+0.0000i 0.0369+0.0000i -0.0202+0.0000i -0.333+0.0000i -0.1839+0.0000i -0.278+0.0000i 0.4791+0.0000i 0+0.0000i
0.2611+0.0000i -0.1354+0.0000i -0.0462+0.0000i -0.2193+0.0000i -0.2669+0.0000i -0.1994+0.0000i -0.1011+0.0000i -0.04+0.0000i 0.1829+0.0000i -0.2745+0.0000i 0.2433+0.0000i -0.0783+0.0000i 0.1598+0.0000i -0.0344+0.0000i 0.3217+0.0000i 0.0569+0.0000i 0.253+0.0000i 0.3438+0.0000i -0.1422+0.0000i -0.4801+0.0000i 0+0.0000i
0.2594+0.0000i -0.173+0.0000i 0.0362+0.0000i -0.1314+0.0000i -0.2244+0.0000i -0.2443+0.0000i -0.2162+0.0000i 0.1536+0.0000i -0.0438+0.0000i -0.1025+0.0000i 0.2299+0.0000i -0.2886+0.0000i -0.2298+0.0000i -0.0563+0.0000i -0.0665+0.0000i -0.1366+0.0000i 0.1931+0.0000i -0.4999+0.0000i 0.3815+0.0000i 0.2085+0.0000i 0+0.0000i
0.2569+0.0000i -0.2036+0.0000i 0.1102+0.0000i -0.0288+0.0000i -0.1272+0.0000i -0.1981+0.0000i -0.2221+0.0000i 0.2429+0.0000i -0.2225+0.0000i 0.151+0.0000i -0.0389+0.0000i -0.0976+0.0000i -0.2275+0.0000i 0.033+0.0000i -0.3002+0.0000i 0.275+0.0000i -0.2906+0.0000i 0.5187+0.0000i -0.1234+0.0000i 0.2125+0.0000i 0+0.0000i
0.2536+0.0000i -0.2273+0.0000i 0.171+0.0000i 0.0716+0.0000i -0.0057+0.0000i -0.0881+0.0000i -0.1303+0.0000i 0.1924+0.0000i -0.2356+0.0000i 0.2586+0.0000i -0.2494+0.0000i 0.2031+0.0000i 0.1126+0.0000i 0.0736+0.0000i 0.0141+0.0000i -0.4544+0.0000i -0.1393+0.0000i -0.2791+0.0000i -0.2949+0.0000i -0.3911+0.0000i 0+0.0000i
0.2496+0.0000i -0.2445+0.0000i 0.2158+0.0000i 0.1561+0.0000i 0.1096+0.0000i 0.0433+0.0000i 0.0094+0.0000i 0.0451+0.0000i -0.0947+0.0000i 0.1516+0.0000i -0.2023+0.0000i 0.2455+0.0000i 0.2727+0.0000i 0.009+0.0000i 0.2874+0.0000i 0.5901+0.0000i 0.275+0.0000i -0.1434+0.0000i 0.2187+0.0000i 0.0927+0.0000i 0+0.0000i
0.2451+0.0000i -0.2554+0.0000i 0.2434+0.0000i 0.2157+0.0000i 0.195+0.0000i 0.1548+0.0000i 0.1372+0.0000i -0.1161+0.0000i 0.0962+0.0000i -0.0645+0.0000i 0.0294+0.0000i 0.0119+0.0000i 0.0617+0.0000i -0.0563+0.0000i 0.0958+0.0000i -0.5431+0.0000i 0.1604+0.0000i 0.4134+0.0000i 0.2317+0.0000i 0.3143+0.0000i 0+0.0000i
0.2401+0.0000i -0.2607+0.0000i 0.2538+0.0000i 0.2458+0.0000i 0.2365+0.0000i 0.2171+0.0000i 0.2075+0.0000i -0.2154+0.0000i 0.221+0.0000i -0.2228+0.0000i 0.2231+0.0000i -0.222+0.0000i -0.2163+0.0000i -0.0338+0.0000i -0.2263+0.0000i 0.2286+0.0000i -0.2265+0.0000i -0.2287+0.0000i -0.2283+0.0000i -0.2288+0.0000i 0+0.0000i
c=
7.1136 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1.0781 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0.3324 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.1746 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0.1063 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0.0762 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.0581 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0.0464 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0.0368 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0.0298 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0.0244 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0.0203 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0.0171 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0.0157 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0144 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0063 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0123 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0076 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0105 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0089 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
好，所以复杂的虚部又在这里(为清楚起见，我将其从矩阵c中删除了)，但请注意.在特征值矩阵c中，再次存在相同的值，但顺序相反.这是为什么?乍一看，似乎b矩阵中的值也相同，只是列被反转了，但是存在一些差异.但据我了解，这是可以的.
Ok, so the complex imaginary part is here again (i removed it from matrix c for clarity) but notice something. In eigenvalue matrix c there are the same values again but in reverse order. Why is that? At first glance is seems that values in b matrices are also the same just columns are reversed, but there are some differences. But as i understand this is ok.
最后
我对男子汉和两件事感到困惑.一种是这种转换为复数，另一种是这种逆序.
I'm confused manly with two things. One is this transformation to complex numbers and the other one is this reverse ordering.
有人对发生的事情有一些见识吗?可以通过更新的MATLAB版本或工具链解决第一种现象吗? (我在Ubuntu 12.04上使用MATLAB 2013，GCC 4.6).第二种现象很可能可以用数学和已实现的算法来解释，以计算特征向量/值.有什么可能的解释吗?
Does anyone have some insight on whats going on. Could first phenomena be resolved with updated MATLAB version or tool-chain? (I'm using MATLAB 2013, GCC 4.6 on Ubuntu 12.04). Second phenomena can probably be explained with math and implemented algorithms to compute eigenvectors/values. Any possible explanation?
推荐答案
对于特征值问题，不仅只有一个有效的答案.每当处理此类情况时，最佳实践就是使用或定义规范形式，以将等效答案转换为相同答案.要将任何答案转换成这样的规范形式，我将执行以下步骤:
There is not only one valid answer for a eigenvalue problem. Whenever dealing with such cases, best practice is to use or define a canonical form to convert equivalent answers to identical answers. To convert any answer into such a canonical form, I would apply the following steps:
- 将所有特征向量标准化为长度1(例如:
[-0.4472,-0.4472,-0.4472,0.4472,0.4472]'而不是[-1,-1,-1,1,1]').可以使用b=bsxfun(@rdivide,b,sqrt(sum(b.^2,1)))实现
- 对于在第一分量中具有负值的每个特征向量，取负值. (例如:
[0.4472,0.4472,0.4472,-0.4472,-0.4472]'而不是[-0.4472,-0.4472,-0.4472,0.4472,0.4472]').可以使用b=bsxfun(@times,sign(b(1,:)),b)来实现
- 以特征值的升序对特征向量和特征值进行排序.可以使用此代码 实现
- Normalize all eigenvectors to length 1 (example:
[-0.4472,-0.4472,-0.4472,0.4472,0.4472]'instead of[-1,-1,-1,1,1]'). Could be achieved usingb=bsxfun(@rdivide,b,sqrt(sum(b.^2,1))) - For each eigenvectors with a negative value in the first component, take the negative value. (example:
[0.4472,0.4472,0.4472,-0.4472,-0.4472]'instead of[-0.4472,-0.4472,-0.4472,0.4472,0.4472]'). Could be achieved usingb=bsxfun(@times,sign(b(1,:)),b) - Sort eigenvectors and eigenvalues in ascending order of the eigenvalues. Could be achieved using this code
请注意，这只是一种规范形式，而不是规范形式.可能建立了规范形式，但是我没有找到任何参考，因此我基本上将上面讨论的内容放在一起定义了规范形式.从数学上讲，应用这样的规范形式可得出独特的解决方案.在实践中，由于浮点精度误差，您会发现细微的差别.
Please note that this is just one canonical form, not the canonical form. There might be canonical forms established but I have not found any reference so I basically put together what we discussed above to define a canonical form. Mathematically, applying such a canonical form results in a unique solution. In practice you will observe minor differences because of floating point precision errors.
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